31-EFT.WP.BH.TensionWall v1.0 | Appendix A — Symbols, Units & Dimensional Checks (TW Edition)

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Appendix A — Symbols, Units & Dimensional Checks (TW Edition)

• I. One-Sentence Aim
  Provide unified notation and SI unit/dimension conventions for symbols related to the Tension Wall Sigma_TW, together with an executable dimensional-check workflow and guards against common pitfalls—so that T_arr, n_eff, Phi_T, and wall terms W(r), Xi_TW(r), Delta_T_sigma remain dimensionally self-consistent under both arrival-time gauges.
• II. Applicability & Non-Goals
• Covered: base dimensions; unified SI units; TW-specific symbols and dimensions; continuous/discrete checks for both gauges; parameter dimensions for wall and band terms; dimensions for interface quantities and the energy triplet; automated checking steps and examples.
• Not covered: device-level unit restrictions; replacements for Chapter 7 uncertainty propagation or Chapter 9 numerical details.
• III. Base Dimensions & SI Norms
• Base dimensions: [L] (length), [T] (time), [M] (mass). Angular quantities are dimensionless by default (radians).
• SI units: length m, time s, speed m·s^-1, frequency Hz = s^-1. Path line element and path length use meters.
• Rules: wrap inline symbols in backticks; parenthesize all divisions, integrals, and composite operators; explicitly declare the path gamma(ell) and the measure d ell.
• IV. Fixed Symbols & Corresponding Dimensions (TW Edition)
• Space, time & path
• x: position, dim = [L]; t: time, dim = [T]
• ell: path parameter, dim = [L]; gamma(ell): path map, dim = [L]
• d ell: line element, dim = [L]; L_path = ∫ d ell, dim = [L]
• Arrival time & speeds
• T_arr: arrival time, dim = [T]
• c_ref: reference speed (constant or field), dim = [L][T^-1]
• c_loc(x,t,f) = c_ref / n_eff, dim = [L][T^-1]
• Refractive index & decomposition
• n_eff(x,t,f): effective refractive index, dim = 1
• n_common(x,t): common term, dim = 1; n_path(x,t,f): path/band term, dim = 1
• Wall & interface
• W(r): normalized wall profile (0→1), dim = 1
• Xi_TW(r) = | dW/dr |, dim = [L^-1]
• r_H: characteristic radius, dim = [L]; Delta_w: wall thickness, dim = [L]
• Delta_T_sigma: zero-thickness correction to arrival time, dim = [T]
• Potential & gradient
• Phi_T(x,t): tension potential, dim = ? (implementation may non-dimensionalize)
• grad_Phi_T(x,t): dim = dim(Phi_T)[L^-1]
• Interface quantities & energy triplet
• C_sigma = Phi_T^+ − Phi_T^-, dim = dim(Phi_T)
• J_sigma = dot( grad_Phi_T^+ − grad_Phi_T^- , n_vec ), dim = dim(grad_Phi_T)
• R_TW, T_trans, A_sigma: reflection, transmission, loss coefficients, dim = 1, with R_TW + T_trans + A_sigma = 1
• Directional vectors
• n_vec: outward unit normal, dim = 1; t_hat: path-unit tangent, dim = 1
• V. Dimensional Checks for the Two Arrival-Time Gauges (Continuous & Discrete)
• Constant-factored gauge
• Continuous: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  Check: ∫ n_eff d ell → [L]; multiply by (1/c_ref) → [T].
• Discrete: T_arr ≈ ( 1 / c_ref ) * ∑ n_eff[ gamma[k] ] · Δell[k] → [T]
• General gauge
• Continuous: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
  Check: (n_eff/c_ref) → [T][L^-1]; times d ell → [T].
• Discrete: T_arr ≈ ∑ ( n_eff[ gamma[k] ] / c_ref[ gamma[k] ] ) · Δell[k] → [T]
• Lower-bound consistency: since n_eff ≥ 1, always T_arr ≥ L_path / c_ref (equivalently embedded in the general-gauge integrand).
• VI. Parameter Dimensions for Wall & Band Terms
• Isotropic small-gradient expansion (Chapter 3 S40-5)
• n_eff ≈ a0 + a1 · ( Phi_T − Phi_0 ) + a2 · norm( grad_Phi_T )^2 + u0 · W(r) + u1 · Xi_TW(r)
• dim(a0) = 1; dim(a1) = 1 / dim(Phi_T)
• dim(a2) = 1 / dim(grad_Phi_T)^2 = [L^2] / dim(Phi_T)^2
• dim(u0) = 1; dim(u1) = 1 / [L]
• Directional extension (Chapter 3 S40-6)
• … + b1 · dot( grad_Phi_T , t_hat ) + b1_sigma · dot( grad_Phi_T , n_vec )
• dim(b1) = dim(b1_sigma) = 1 / dim(grad_Phi_T) = [L] / dim(Phi_T)
• Band polynomial (Chapter 3 S40-7)
• n_path ≈ ∑_{m=1}^M c_m(x,t) · ( f − f0 )^m, with dim( (f − f0)^m ) = [T^-m]
• dim(c_m) = [T^m]; if using normalized ((f − f0)/f_ref)^m, then dim(c_m) = 1
• Zero-thickness correction (Chapter 3 S40-11)
• Delta_T_sigma ≈ k_sigma · H( crossing ), with dim(H) = 1, hence dim(k_sigma) = [T]
• VII. Interface & Inter-Layer Dimensional Checks
• Continuous: Phi_T^+ = Phi_T^-, J_sigma = 0; if F is continuous, then n_eff^+ = n_eff^- (dimensionally coherent).
• Potential jump: C_sigma ≠ 0, dim(C_sigma) = dim(Phi_T); effects on n_eff propagate through a1, b1/b1_sigma dimensions.
• Flux jump: J_sigma ≠ 0, dim(J_sigma) = dim(grad_Phi_T); if a wall-normal response coefficient k_sigma^n is added in n_eff, then dim(k_sigma^n) = 1 / dim(grad_Phi_T).
• Feasible side limits: after matching, enforce n_eff^+ ≥ 1 and n_eff^- ≥ 1; any violation is infeasible.
• VIII. Uncertainties & Dimensions
• Rule: u(q) carries the same dimensions as q; combined u_c(T_arr) has [T].
• Typical items: u(c_ref) → [L][T^-1], u(n_eff) → 1, u(Δell) → [L], u(T_arr) → [T], u(Delta_T_sigma) → [T].
• Reporting: always use mean ± k·u_c in [T]; the differential ΔT_arr(f1,f2) is also [T].
• IX. Coordinates & Unit Mappings (Implementation Contract)
• units_spec: declare length m, time s, speed m·s^-1, frequency Hz. If inputs come in km or ms, convert to SI at ingress and record the mapping.
• coords_spec: declare the coordinate frame (Cartesian/Spherical or ECEF/ENU); any transform must preserve the units of Δell and update hashes.
• Path constraints: gamma[k] and Δell[k] share units; t_hat[k] and n_vec are dimensionless; interface coordinates use the same frame as the path.
• X. Automated Dimensional-Check Workflow (check_dimension recommended order)
• Contract intake: load units_spec, coords_spec, mode, thresholds eta_T, eta_w, tau_switch.
• Symbol binding: fix dim(c_ref)=[L][T^-1], dim(n_eff)=1, dim(d ell)=[L], dim(T_arr)=[T].
• Equation checks:
• Constant-factored: verify dim( ∫ n_eff d ell / c_ref ) = [T]
• General: verify dim( ∫ ( n_eff/c_ref ) d ell ) = [T]
• TW-term checks: infer and validate dimensions for a1, a2, u1, b1, b1_sigma, c_m, k_sigma against this appendix.
• Interface-term checks: ensure C_sigma, J_sigma dimensions are coherent with their effects in n_eff; R_TW + T_trans + A_sigma is an energy balance (dimensionless).
• Discrete consistency: confirm units for Δell[k] and c_ref[k] match; include segment endpoints explicitly; forbid cross-interface interpolation.
• Report: emit DimReport (see Section XII) and write to logs.
• XI. Common Errors & Safeguards
• Unit mixing: path in km while c_ref in m·s^-1 → T_arr off by 10^3. Safeguard: normalize at ingress and record.
• Missing parentheses: ∫ n_eff / c_ref d ell vs ∫ ( n_eff / c_ref ) d ell are not equivalent. Safeguard: enforce parentheses.
• Angle units: passing degrees to trig functions. Safeguard: declare radians and convert if needed.
• Parameter mis-dimensioning: c_m not [T^m] or u1 not [L^-1]. Safeguard: infer and validate during model assembly.
• Interface mislabeling: entering C_sigma or J_sigma as dimensionless. Safeguard: correct to dim(Phi_T) / dim(grad_Phi_T) or reject.
• Naming confusion: never mix T_fil with T_trans; never mix n (number density) with n_eff (effective index).
• XII. Minimal DimReport Fields (TW Edition)
• units_spec, coords_spec, mode ∈ {constant, general}
• ok(formula_S40-10), ok(lower_bound), ok(discrete_forms)
• dims: { c_ref:[L T^-1], n_eff:1, d_ell:[L], T_arr:[T], Phi_T:?, grad_Phi_T:?·[L^-1], W:1, Xi_TW:[L^-1] }
• params_dims: { a0:1, a1:1/dim(Phi_T), a2:[L^2]/dim(Phi_T)^2, u0:1, u1:[L^-1], b1:[L]/dim(Phi_T), b1_sigma:[L]/dim(Phi_T), c_m:[T^m], k_sigma:[T] }
• interfaces: { C_sigma:dim(Phi_T), J_sigma:dim(grad_Phi_T), R_TW:1, T_trans:1, A_sigma:1 }
• tw_switch: { eta_w:1, tau_switch:[T] }
• discrete_consistency: { Δell_unit:ok, c_ref_unit:ok, segmentation:no-cross-interface, interpolation:ok }
• notes: anomalies & auto-fix notes
• hashes: hash(Phi_T), hash(gamma), hash(code), hash(TWProfile)
• XIII. Worked Checks (Four Examples)
• Example 1 | Uniform outer region, no wall
• Condition: n_eff ≡ 1, arbitrary path.
• Conclusion: T_arr = L_path / c_ref, dimension [L]/([L][T^-1]) = [T]; lower bound is tight.
• Example 2 | Thin-wall zero-thickness correction
• Condition: single crossing, Delta_T_sigma = k_sigma.
• Dimension: dim(k_sigma) = [T]; T_arr_total = T_arr + Delta_T_sigma remains [T].
• Example 3 | First-order band dispersion
• Condition: n_path ≈ c_1 ( f − f0 ).
• Dimension: dim(c_1) = [T], hence dim(ΔT_arr) = [T]; both gauges agree.
• Example 4 | Directional terms (normal/tangential)
• Condition: n_eff ≈ a0 + b1 · dot( grad_Phi_T , t_hat ) + b1_sigma · dot( grad_Phi_T , n_vec ).
• Dimensions: dim(b1) = dim(b1_sigma) = [L]/dim(Phi_T); if Phi_T is non-dimensionalized, dim(b1) = dim(b1_sigma) = [L].
• XIV. Cross-References
• EFT.WP.BH.TensionWall v1.0 Chapter 3 (Minimal Equations & Structural Model), Chapter 5 (Wall Parameterization), Chapter 6 (Propagation & Arrival Time), Chapter 8 (Interface Matching), Chapter 12 (Error Budget)
• EFT.WP.Propagation.TensionPotential v1.0 Appendix A (Symbols & Dimensions)
• EFT.WP.Core.Equations v1.1 S06-* (notation & operators)
• EFT.WP.Core.Metrology v1.0 M05-, M10- (units & traceability)
• XV. Deliverables
• Dimension–unit binding checklist (ready to embed in the Contract).
• Execution order and decision rules for check_dimension (including TW terms).
• Sample DimReport and log-field mapping (including tau_switch and energy-triplet entries).